Chapter 1. Definition and elementary properties of triangulated categories

In this paper we introduce an algebra embedding $\iota:K \to S$ from the free associative algebra $K $ generated by a finite or countable set $X$ into the skew monoid ring $S = P * \Sigma$ defined by the commutative polynomial ring $P = K[X\times N^*]$ and by the monoid $\Sigma = $ generated by a suitable endomorphism $\sigma:P\to P$. If $P = K[X]$ is any ring of polynomials in a countable set of commuting variables, we present also a general Gr\"obner bases theory for graded two-sided ideals of the graded algebra $S = \bigoplus_i S_i$ with $S_i = P \sigma^i$ and $\sigma:P \to P$ an abstract endomorphism satisfying compatibility conditions with ordering and divisibility of the monomials of $P$. Moreover, using a suitable grading for the algebra $P$ compatible with the action of $\Sigma$, we obtain a bijective correspondence, preserving Gr\"obner bases, between graded $\Sigma$-invariant ideals of $P$ and a class of graded two-sided ideals of $S$. By means of the embedding $\iota$ this results in the unification, in the graded case, of the Gr\"obner bases theories for commutative and non-commutative polynomial rings. Finally, since the ring of ordinary difference polynomials $P = K[X\times N]$ fits the proposed theory one obtains that, with respect to a suitable grading, the Gr\"obner bases of finitely generated graded ordinary difference ideals can be computed also in the operators ring $S$ and in a finite number of steps up to some fixed degree.

Skew polynomial rings, Groebner bases and the letterplace embedding of the free associative algebra

We introduce and study potentials, mutations and Jacobian algebras in the framework of tensor algebras associated with symmetrizable dualizing pairs of bimodules on a symmetric algebra over any commutative ground ring. The graded context is also considered by starting from graded bimodules, and the classical non simply-laced context of modulated quivers with potentials is a particular case. The study of potentials in this framework is related to symmetrically separable algebras, and we have two kinds of potentials: the symmetric and the non symmetric ones. When the Casimir ideal of the symmetric algebra coincides with its center, all potentials appear as symmetric potentials and their manipulation mimics the simply laced study of quivers with potentials. This useful information suggests that, for applications to cluster algebras theory and related fields, one may restrict a further study of modulated quivers with potentials to the setting where the ground symmetric algebra is separable over a field. Associated with this work is a generalized construction of Ginzburg dg-algebras and cluster categories associated with graded modulated quivers with potentials.

Potentials and Jacobian algebras for tensor algebras of bimodules

/pdf/hopf-algebras-quantum-groups-and-yang-baxter-equations-hscvsqo410.pdf

Hopf Algebras, Quantum Groups and Yang-Baxter Equations

Zacharias representation of effective associative rings

Effective Buchberger-Zacharias-Weispfenning theory of skew polynomial extensions of subbilateral coherent rings

A non-simply-laced version for cluster structures on 2-Calabi–Yau categories

This work formally introduces and starts investigating the structure of matrix polynomial algebra extensions 
of a coefficient algebra by (elementary) matrix-variables over 
 a ground polynomial ring in not necessary commuting variables. 
 These matrix subalgebras of full matrix rings over polynomial rings show up 
 in noncommutative algebraic geometry. We carefully study their (one-sided or bilateral) noetherianity, obtaining a precise lift of the Hilbert Basis Theorem when the 
ground ring is either a commutative polynomial ring, a free noncommutative polynomial ring or a skew polynomial ring extension by a free commutative term-ordered monoid. 
We equally address the natural but rather delicate question of recognising which matrix polynomial algebras are Cayley-Hamilton algebras, 
which are interesting noncommutative algebras arising from the study of $\mathrm{Gl}_{n}$-varieties.

/pdf/the-structure-of-matrix-polynomial-algebras-14ybigg0.pdf

The structure of matrix polynomial algebras

The noncommutative geometry of matrix polynomial algebras

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Bertrand Nguefack

Author Tools

Chat about Author

Papers

Effective Buchberger-Zacharias-Weispfenning theory of skew polynomial extensions of subbilateral coherent rings

A non-simply-laced version for cluster structures on 2-Calabi–Yau categories

The structure of matrix polynomial algebras

The noncommutative geometry of matrix polynomial algebras